There is no direct conversion of graphite to carbon fibre which is practical on a large scale, though graphite "whiskers" can be produced this way on a very small scale -- more of a laboratory technique than a process suitable for production.
Carbon "nanofibres" can be grown from hydrocarbon precursors such as methane, and methane can be made from CO2 and H2. So if the H2 is "green", you've got a way to convert CO2 to CH4 to carbon fibre, but the conversion will be many, many times slower than the production of CO2, and consume quite a bit of energy as well.
Currently, carbon fiber is made mostly by pyrolysis (high-temperature decomposition) of carbon-containing precursors such as polyacrylonitrile. Acrylonitrile is made from propene, ammonia, and oxygen, with propene coming mostly form petroleum, so either one needs to find a competing process, or find a way to convert CO2 to propene. Some propene has been made from methanol, but I don't have details of this (it's probably safe to assume it's the methanol-to-gasoline process invented by Mobil a few decades ago) -- presumably a mixture of products is formed.
It's quite conceivable one could obtain propene from agricultural byproducts (which are just captured CO2) such as acetone (obtained by starch fermentation); but currently it is cheaper to go the other direction, and make acetone from propene, because of the artificially low cost of heavily-subsidised petroleum.
IF we can recover the CO2 economically, and IF we can obtain enough H2 in a renewable and economically competitive fashion, then CO, methane and methanol derived from CO2 will become just another stream of simple precursors to feed into the process streams built around inputs of coal, natural gas, and petroleum. The problem at this point is that the deck is stacked in favor of petroleum-based inputs.
Bear in mind that only a few percent of petroleum goes to make materials -- the majority is burned as fuel. So even industrial-scale conversion of CO2 to materials would only offset a tiny part of CO2 production from burning fossil fuels. Probably it would be better to convert CO2 into fuels such as methanol or dimethyl ether, which would at least form a closed carbon cycle when burned, without the need to bring more fossil carbon to the surface.
Also, if you're thinking of building "big stuff", keep in mind that carbon fiber is combustible -- you might want to think twice before using it in buildings, bridges, etc. in place of non-flammable steel and concrete.
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